Efficient simulation of quantum systems driven by diffusive Markov processes.

Mohan Sarovar, Matthew Grace

Mohan Sarovar, Matthew Grace

The expansion of a stochastic Liouville equation for the coupled evolution of a quantum system and an Ornstein-Uhlenbeck process into a hierarchy of coupled differential equations is a useful technique that simplifies the simulation of stochastically driven quantum systems. We expand the applicability of this technique by completely characterizing the class of diffusive Markov processes for which a useful hierarchy of equations can be derived. The expansion of this technique enables the examination of quantum systems driven by non-Gaussian stochastic processes with bounded range. We present an application of this extended technique by simulating Stark-tuned Förster resonance transfer in Rydberg atoms with nonperturbative position fluctuations.

Clément Pellegrini

"Quantum trajectories" are solutions of stochastic differential equations also called Belavkin or Stochastic Schrödinger Equations. They describe random phenomena in quantum measurement theory. Two types of such equations are usually considered, one is driven by a one-dimensional Brownian motion and the other is driven by a counting process. In this article, we present a way to obtain more advanced models which use jump-diffusion stochastic differential equations. Such models come from solutions of martingale problems for infinitesimal generators. These generators are obtained from the limit of generators of classical Markov chains which describe discrete models of quantum trajectories. Furthermore, stochastic models of jump-diffusion equations are physically justified by proving that their solutions can be obtained as the limit of the discrete trajectories.

Mads Peter Soerensen

David M. Ceperley

Heinz‐Peter Breuer, Francesco Petruccione

Abstract The formulation of the dynamics of open quantum systems by means of stochastic processes in Hilbert space leads to efficient Monte–Carlo simulation techniques which are introduced and examined in this chapter. Depending on whether the stochastic process is a piecewise deterministic process or a diffusion process, the corresponding individual realizations consist of intervals of deterministic evolution periods interrupted by instantaneous quantum jumps, or of continuous, nowhere differentiable paths. Various Monte Carlo algorithms for both types of processes are described in detail, and their convergence behaviour and their numerical performance is investigated for a number of applications, such as the damped harmonic oscillator, the driven two-level system, and the damped driven Morse oscillator.